Ayazoglu, Rabil

We study the quasilinear attraction–repulsion chemotaxis system of parabolic–elliptic type with logistic source involving the exponents depending on the spatial variables: ut=Δu-χ∇·uu+1r-1∇υ+ξ∇·uu+1r-1∇ω+au-bum(x), 0=Δυ-βυ+αu, 0=Δω-δω+γu, where α, β, δ, γ, χ, ξ, b>0, a≥0, r∈R and m:Ω→1,∞ is a measurable function, subject to the homogeneous Neumann boundary conditions in a bounded domain RNN≥1 with smooth boundary. We prove that this system possesses a unique global bounded classical solution, which is an extension of known results, if the repulsion cancels the attraction in the sense that (balance) χα=ξγ with essinfx∈Ωmx>r+N-2+N,1, and if the attraction prevails over the repulsion in the sense that χα>ξγ with essinfx∈Ωmx>maxr+1,1, and if the repulsion prevails over the attraction in the sense that χα<ξγ.

AtıfScopus2·WoS2

İncele

Ayazoglu, Rabil; Akkoyunlu, Ebubekir

This paper deals with the higher dimension quasilinear parabolic–parabolic chemotaxis model involving a source term of logistic type ut= ∇ · (ϕ(u) ∇ u) - ∇ · (ψ(u) ∇ υ) + g(x, u) , τυt= Δ υ- υ+ u, in(x,t)∈Ω×(0,T), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is a smooth and bounded domain in RN, N≥ 1 and ψ, ϕ, g are smooth, positive functions satisfying νsq≤ ψ≤ χsq, ϕ≥ σsp, p, q∈ R, ν, χ, σ> 0 when s≥ s> 1 , g(x, s) ≤ ηsk(x)- μsm(x) for s> 0 , η≥ 0 , μ> 0 constants and g(x, 0) ≥ 0 , x∈ Ω , where k, m are measurable functions with 0≤k-:=essinfx∈Ωk(x)≤k(x)≤m+:=esssupk(x)x∈Ω<+∞, 1<m-:=essinfx∈Ωm(x)≤m(x)≤m+:=esssupm(x)x∈Ω<+∞. We extend the constant exponents k= { 0 , 1 } , m> 1 which in logistic source term g(s) ≤ ηsk- μsm for s> 0 , η≥ 0 , μ> 0 as variable exponents k(·) ≥ 0 , m(·) > 1 with k+< m-. It is proved that if q= m-- 1 (critical case) with μ properly large that μ > μ for some μ> 0 , then there exists a classical solution which is global in time and bounded. Furthermore, if q< m-- 1 , we prove that the classical solutions to the above system are uniformly in-time-bounded without restriction on μ.

AtıfScopus10·WoS8·OpenAlex10

İncele